Permutation Representations Research Articles

The Terwilliger algebras of the group association schemes of three metacyclic groups

AbstractFor any finite group , the Terwilliger algebra of the group association scheme satisfies the following inclusions: , where is a specific vector space and is the centralizer algebra of the permutation representation of induced by the action of conjugation. The group is said to be triply transitive if . In this paper, we determine the dimensions of and for being , and , and show that and are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of , and when they are triply transitive.

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

ON STRONGLY BASE-TWO FINITE GROUPS WITH TRIVIAL FRATTINI SUBGROUP: CONJUGACY CLASSES AND CORE-FREE SUBGROUP

This paper explores finite groups G with a focus on those that are strongly base-two and possess a trivial Frattini subgroup. The concept of base size, denoted by b(G, H), for the action of G on core-free subgroups H, plays acrucial role. The paper investigates the number of conjugacy classes of core-free subgroups with base size exceeding 3, denoted by α(G). A group is considered strongly base-two if α(G) ≤ 1, indicating that nearly all faithful transitive permutation representations of G exhibit a base size of 2. The study delves into the characterization of such groups, shedding light on their properties and structures.

Composing group action to permutation representation

A mapping that satisfies two specific axioms provides a common notion of group action. A homomorphism translating from a group to a symmetric group of a certain set can also be used to describe group action. Therefore, any example of the group actions can be stated based on the second equivalent definition, such as the regular action, natural matrix action, coset action, and Z^2 acting on R^2, etc. It is necessary to examine the concepts of the orbit and stabilizer of a group in order to reveal the orbit-stabilizer theorem. After the preparatory work, the orbit-stabilizer theorem can be proved by defining a mapping from the orbit to the stabilizer and then checking that the mapping is well-defined and bijective. To derive Burnsides lemma, it needs to introduce the set of fixed points which is related to the concept of the stabilizer. Through the orbit-stabilizer theorem along with the fact that a set is a disjoint union of orbits, Burnside's lemma can be confirmed. Moreover, it is natural to compose a group action with a linear representation, and then a representation would be obtained, which is permutation representation. Further, one must calculate the character of the permutation representation, the dimension of the fixed subspace, and the dimension of CX^G. Then it can show Burnsides lemma in another way by permutation representation.

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Representations on the cohomology of [formula omitted

The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1[n] of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.

Induced permutations for approximate metric search

Permutation-based Indexing (PBI) approaches have been proven to be particularly effective for conducting large-scale approximate metric searching. These methods rely on the idea of transforming the original metric objects into permutation representations, which can be efficiently indexed using data structures such as inverted files.The standard conceptualization of permutation associated with a metric object involves only the use of object distances and their relative orders from a set of anchors called pivots. In this paper, we generalized this definition in order to enlarge the class of permutation representations that can be used by PBI approaches. In particular, we introduced the concept of permutation induced by a space transformation and a sorting function, and we investigated which properties these transformations should possess to produce permutations that are effective for metric search. Furthermore, as a practical outcome, we defined a new type of permutation representation that is calculated using distances from pairs of pivots. This proposed technique allowed us to produce longer permutations than traditional ones for the same number of object-pivot distance calculations. The advantage lies in the fact that when longer permutations are employed, the use of inverted files built on permutation prefixes leads to greater efficiency in the search phase.

Phase retrieval for affine groups over prime fields

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group Zp⋊Zp⁎ (with p prime), acting on the vectors in Cp with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimal degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimal faithful degree of the opposite monoid of the full transformation monoid.

The Number of Occurrences of Patterns in a Random Tree or Forest Permutation

The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of these random permutations. The main results show that the distributions of these numbers are asymptotically normal.
 The proof uses representations of random tree and forest permutations that enable us to express the number of occurrences of a pattern by a type of U-statistics; we then use general limit theorems for the latter.

Relative homology of invariant group chains

Let Q be a finite group acting by automorphisms on a discrete group G. K. Knudson has introduced the homology of invariant group chains H⁎Q(G,A). A useful tool in homology is its relative version. In this paper we introduce the homology of Q-permutation representations as a generalization of the homology of permutation representations defined by Snapper. We prove the existence of a spectral sequence which converges to the homology of a Q-permutation representation. We also introduce a relative version of the homology of invariant group chains using the notion of Q-permutation representations following Adamson and give a topological interpretation of that relative homology.

Strongly Base-Two Groups

Let G be a finite group, let H be a core-free subgroup and let b(G, H) denote the base size for the action of G on G/H. Let alpha (G) be the number of conjugacy classes of core-free subgroups H of G with b(G,H) ge 3. We say that G is a strongly base-two group if alpha (G) le 1, which means that almost every faithful transitive permutation representation of G has base size 2. In this paper, we study the strongly base-two finite groups with trivial Frattini subgroup.

Distance and adjacency spectra and eigenspaces for three (di)graph lifts: A unified approach

By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are a complement of Dalfó et al. (2021) [6] and Deng et al. (2007) [8].

Symmetric polynomials over finite fields

It is shown that two vectors with coordinates in the finite q-element field of characteristic p belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree pk,2pk,…,(q−1)pk, k=0,1,2,… has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when q=p and the dimension is large compared to p. A relatively small separating set of multisymmetric polynomials over the field of q elements is derived.

Totally 2-closed finite groups with trivial Fitting subgroup

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Cubic graphical regular representations of Ree groups

A graphical regular representation of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper, we prove that for Ree groups with q > 3, with probability tending to 1 as , a random involution y together with a fixed element x with order q – 1 gives rise to a cubic graphical regular representation of . A similar result involving a fixed element with order is also proved with the help of certain properties of given in [Leemans, D. Liebeck, M. W. (2017). Chiral polyhedra and finite simple groups. Bull. London Math. Soc. 49: 581–592].

Lie Elements and the Matrix-Tree Theorem

For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) \subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original representation V. Lie elements often exhibit nice combinatorial properties. Thus, for G = S_n and V, a permutation representation, we prove a formula for the characteristic polynomial of a Lie element similar to the classical matrix-tree theorem.

Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-theory

In our previous work, we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko’s results on permutation representations.

Adaptive Evolutionary Multitasking to Solve Inter-Domain Path Computation Under Node-Defined Domain Uniqueness Constraint: New Solution Encoding Scheme

In multi-domain networks, the efficiency of path computation becomes more and more important. The Inter-Domain Path Computation under Node-defined Domain Uniqueness Constraint (IDPC-NDU) is a recently investigated problem where its objective is to determine the effective routing path between two nodes that traverses every domain at most once. IDPC-NDU is NP-Hard, so the approximation approaches are suitable to deal with this problem for large instances. Multifactorial Evolutionary Algorithm (MFEA) is an emerging research topic in the field of evolutionary computation that can efficiently tackle multiple optimization problems at the same time. This study proposed an approach based on the combination of the Adaptive Multifactorial Evolutionary Algorithm (dMFEA-II) and Dijkstra algorithm for solving IDPC-NDU. The encoding and evaluating methods based on the permutation representation are also introduced, and the new individual representation is always to produce valid solutions. The proposed algorithm is evaluated on two types of instances. Simulation results demonstrate the superior performance of the proposed algorithm in comparison with the existing algorithms in terms of the quality of the solution.

Algorithm for Combinatorial Bipartite Enumeration of Chiral and Achiral Stereoisomers of Homo- and Heteropolysubstituted Derivatives of the Cubane Molecule

This paper is devoted to the application of an algorithm for the enumeration of chiral and achiral stereoisomers of homo and heteropolysubstituted derivatives of the cubane molecule. The sequence of this algorithm is in three steps, the first of which is the determination of the permutations of 8 hydrogen atoms of the cuban molecule ( ) subjected to the action of the Oh group. The second step will be the composition of these permutations later into permutational representations and controlling respectively the chirality and achirality of this molecular system. The transformation of these 2 operators into a pair of generic formulae used for bipartite combinatorial enumeration giving chiral and achiral isomer numbers of the homo- and hetero-polysubstituted derivatives of the cubane molecule respectively symbolised by the empirical formulae and being the final step.